Paper “Normalisers of residuals of finite groups” published in Arch. Math. (Basel)

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A. Ballester-Bolinches, S. F. Kamornikov, and H. Meng

Normalisers of residuals of finite groups

Arch. Math. (Basel), 109(4):305–310, 2017

https://doi.org/10.1007/s00013-017-1074-8

Abstract:

Let F be a subgroup-closed saturated formation of finite groups containing all finite nilpotent groups, and let M be a subgroup of a finite group G normalising the F-residual of every non-subnormal subgroup of G. We show that M normalises the F-residual of every subgroup of G. This answers a question posed by Gong and Isaacs (Arch Math 108:1–7, 2017) when F is the formation of all finite supersoluble groups.

2010 Mathematics Subject Classification: 20D10, 20D35

Keywords: Finite group, Formation, Residual, Subnormality

Paper “Finite groups with all minimal subgroups solitary” published in J. Algebra Appl.

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R. Esteban-Romero and Orieta Liriano.

Finite groups with all minimal subgroups solitary.

J. Algebra Appl., 15(8):1650140, 9, 2016

https://doi.org/10.1142/S0219498816501401

Abstract 

We give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order.

2010 Mathematical Subject Classification: 20D10, 20D30

Keywords: Finite group; solitary subgroup; minimal subgroup

Paper “On locally finite groups whose subgroups of infinite rank have some permutable property” published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, S. Camp-Mora, M. R. Dixon, R. Ialenti, and F. Spagnuolo

On locally finite groups whose subgroups of infinite rank have some permutable property

Ann. Mat. Pura Appl. (4), 196(5):1855–1862, 2017

https://doi.org/10.1007/s10231-017-0642-7

Abstract

In this paper, we study the behavior of locally finite groups of infinite rank whose proper subgroups of infinite rank have one of the three following properties, which are generalizations of permutability: S-permutability, semipermutability and S-semipermutability. In particular, it is proved that if G is a locally finite group of infinite rank whose proper subgroups of infinite rank are S-permutable (resp. semipermutable), then G is locally nilpotent (resp. all subgroups are semipermutable). For locally finite groups whose proper subgroups of infinite rank are S-semipermutable, the same statement can be proved only for groups with min-p for every prime p. A counterexample is given for the general case.

2010 Mathematical Subject Classification: 20F19, 20F50

Keywords: Locally finite group, Section p-rank, Section rank, Special rank, Permutable, Sylow permutable, Semipermutable, S-semipermutable

 

Paper “Some Local Properties Defining T₀-Groups and Related Classes of Groups” published in Publ. Mat.

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A. Ballester-Bolinches, J. C. Beidleman, R. Esteban-Romero, and M. F. Ragland

Some local properties defining T0-groups and related classes of groups

Publ. Mat., 60(1):265–272, 2016

http://projecteuclid.org/euclid.pm/1450818490

Abstract

We call G a Hall_χ-group if there exists a normal nilpotent subgroup N of G for which G/N is an χ-group. We call G a T-group provided G/Φ(G) is a T-group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define Hall_χ-groups and T-groups where χ{T, PT, PST}; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.

2010 Mathematical Subject Classification: 20D10, 20D20, 20D35

Keywords: Subnormal subgroup, T-group, PST-group, finite solvable group